At the level of approximation being used, the sun and the moon have the same apparent size in the sky, as during a total solar eclipse the moon just covers the sun. This being the case, the ratio of their diameters would be the same as the ratio of their distances, and the ratio of their volumes would be the cube of this.

390^3 = 59,319,000

so the ratio of their volumes would be about 59 million to 1.

The mean distance of the earth from the sun is 149,598,261 km, while the moon's averate distance is 384,748 km (the semi-major axes of the respective relevant orbits, as given by Wikipedia), for a ratio of 388.8, so the sun is 387.8 times farther than the moon (388.8 times as far as the moon--note the distinction). That gives a volume ratio of 58,782,834 (using 388.821413^3, as a more exact ratio of the distances, cubed).

Google shows the moon's diameter as 3476 km and the sun's as 1,392,000 Km for a ratio of volumes as 64,220,000 to 4 significant digits. The discrepancy is due to the fact that on average the moon's apparent size in the sky is slightly smaller than the sun's, rather than the same. There are more annular (ring) eclipses of the sun, where a complete ring of bright solar surface is visible around the moon, than total eclipses.